She got her period during Hajj and she cannot stay

She should travel with him and remain in ihraam, then she should come back when she has become pure. This applies if she lives in the land of the Two Holy Places, because it is easy to come back and does not involve a great deal of trouble or need a passport etc. But if she is a foreigner and it is difficult for her to come back, then she should tie a piece of cloth over her private part (so that no blood will leak and contaminate the Mosque), and do tawaaf and saa’i, and cut her hair, and end her ‘Umrah on the same journey, because in this case her tawaaf has become necessary, and in cases of necessity things that are ordinarily forbidden are allowed. 

But she does not have to do tawaaf al-wadaa’ (farewell tawaaf), because menstruating women do not have to do the farewell tawaaf, because of the hadeeth of Ibn ‘Abbaas (may Allaah be pleased with him), that the Prophet (peace and blessings of Allaah be upon him) told the people that the last thing they should do in Makkah was to circumambulate the Ka’bah, but he exempted menstruating women. 

And when the Prophet (peace and blessings of Allaah be upon him) was told that Safiyyah had done tawaaf al-ifaadah, he said, “Then let her leave (for her Hajj is over).” This indicates that the farewell tawaaf is not obligatory for menstruating women, but tawaaf al-ifaadah is essential. 

See the fatwa of Shaykh Ibn ‘Uthaymeen, Essay 60, Questions about menstruation.